Xiomin YANG, Hisheng YANG, Yongcun CUI, Yn LI,Bingzhen JIANG, Sier DENG,*

a School of Mechatronics Engineering, Henan University of Science and Technology, Luoyang 471003, China

b College of Ocean Engineering, Guilin University of Electronic Technology, Beihai 536000, China

KEYWORDS Elastic ring squeeze film damper;Orifice flow;Rolling bearing;Thin-walled ring;Turbulent jet;Vibration

Abstract This paper explores an analytical model for Elastic Ring Squeeze Film Damper(ERSFD) with thin-walled ring and turbulent-jet orifices, and uncovers its Oil Film Pressure Performance (OFPP). Firstly,the ring deformation is addressed by using the Fourier series expansion approach and the orifice outflow rate is characterized with the Prandtl boundary layer theory.Secondly,applying finite difference scheme,the influence of elastic ring flexibility,orifice diameter,and attitude angle on the OFPP is analyzed.Finally,Outer chamber pressure was measured experimentally at different rotor speeds.The results indicate that the outer chamber pressure coats an individual load-carrying region and spreads symmetrically pertaining to the attitude angle. Its amplitude drops as the elastic ring flexibility decreases but boosts with the reduction of the orifice diameter.For inner chamber pressure,the orifice diameter effects a similar trend to the outer cavity,but exhibits more stable distribution regarding the attitude angle. Minimizing the elastic ring flexibility causes an increase in amplitude. The model is validated by the test results giving that the outer chamber pressure shifts synchronously and periodically with the variation of the attitude angle,while the pressure amplitude increases slightly at higher rotor speeds.

1. Introduction

Elastic Ring Squeeze Film Damper (ERSFD) is commonly positioned in military aircraft engines to attenuate high dynamic vibrations, enjoying excellent bearing capacity and damping characteristics under extreme working conditions.1–5 In order to meet the utmost desire of adequate criticalspeed margin and high-stable rotor speed in modern aeroengine, the journal normally whirls at a large orbital radius.In this case,the oil film gap-fixed Squeeze Film Damper(SFD)fails to dampen the rotor vibration due to the rapid nonlinear growth of the radial oil film force. Therefore, a deformable elastic ring is enveloped in the SFD assembly. The rapid increase of radial oil film force at a large orbital radius can be adequately lessened by the expansion deformation of elastic ring when the rotor operating at high speeds.Correspondingly,the insufficient squeezing effect is timely prevented by the recovery deformation of elastic ring when the rotor runs at low speeds. Therefore, it is of a tremendous requirement to study the effect of elastic ring deformation on the Oil Film Pressure Performance (OFPP) for covering vibrations when the rotor traversing multi-critical speeds. Furthermore, the journal experiences heavy transient oscillations when the rotor runs in the state such as engine transition,maneuvering,or flying in a distorted flow field.The journal hits the elastic ring as well squeezes the oil film, so that an impact effect occurs and excess energy is introduced. This shortens the fatigue life of the elastic ring and makes the damper unstable. Therefore, a series of damping orifices are drilled on the elastic ring.It provides the necessary damping force with the journal to dissipate any energy generated by high-speed dynamic motion. This makes the accurate prediction of the oil film pressure even more involved because the instability of orifice flow further interferes with the complex fluid–solid coupling fields. Unfortunately, there is still no accurate mathematical model for ERSFD.6Up to now, finite element technique7–9has been developed to simulate elastic ring deformation based on constructing a fluid–solid coupling model with regular analysis elements,such as thick plate element,10shell element,11and threedimensional entity elements,12etc. This approach benefits accurate pressure results and facilitates dealing with variable operating conditions and complex assembly properties. However, there are some troubles in establishing mathematical models and deriving general laws.13Chen et al.14recently presented an analytical model for calculating the elastic ring radial displacement with an assumption of circular deformation. It concludes that the elastic ring deformation can adequately suppress oil film nonlinearity at large eccentricity. Cao et al.15used a simply-supported beam model to solve the elastic ring deformation according to the theory of material mechanics.However, developing analytical model for elastic ring deformation remains rather obscure due to the lack of good comprehensive understanding of the mechanics such as structural mechanics, heat transfer and fluid mechanics.

On the other side, there is still no definite orifice analytical model in ERSFD application to date. In hydraulic restrictor,the damping orifice commonly has a ratio of length-diameter greater than 4, and its mathematical model has been extensively developed.16–18However, in ERSFD modeling, the diameter of the orifice depends on the high-cycle fatigue limit of the elastic ring,15which is typically not less than the thickness of the elastic ring and performs a characteristic like short hole.The difference in structure and flow characteristics in the two cases makes the hydraulic orifice model not suitable for the ERSFD model.Zhou and Li19proposed a thin-walled orifice model to estimate the net outflow rate through the orifice by satisfying continuity of the flow. Jiang et al.20used finite element approach to assess the flow fields at the local crosssection about the orifice. It suggests that the orifice outflow is a complex process characterized by energy dissipation,velocity discontinuity,and turbulent vortex.The finite element approach is also frequently used to optimize the orifice distribution on the elastic ring,21even for air throttling orifice model,22it illustrates that the orifice diameter has a significant effect on the damping characteristics. However, the velocity profile of the fluid submerged through the orifice is discontinuous, unstable, and accompanied by viscous turbulence.Applying an orifice outflow model suitable for ideal fluid to the ERSFD model is no longer justifiable.

To advance the mathematical model research on ERSFD,this paper proposes a thin-walled ring and turbulent-jet orifice analytical model and investigates the OFPP derived from the Reynolds equations. The elastic ring deformation and orifice net outflow rate are obtained respectively following the planar bending theory of thin-walled ring and the Prandtl boundary layer differential equations. Apply central finite difference scheme to solve the pressure governing equations,and perform numerical analysis to study the effect of elastic ring flexibility,orifice diameter, and attitude angular location on the OFPP.Experiment was carried out on a High Impact & Overload Integrated Bearing Fatigue Testing Machine to verify the proposed model by measuring the pressure distribution of the outer cavity at different rotor speeds. The proposed analytical model advances for calculating the pressure field of the ERSFD located on rolling bearings.

2. Model considerations and solving algorithm

2.1. Oil film pressure equations

Fig. 1 schematically shows a single-layer ERSFD model. It mainly consists of a journal of radius R, two seal rings, and a nonrotating elastic ring inlayed between the inside and exterior wall.The elastic ring is circumferentially positioned by its first inner pedestal angle β relative to the-Y coordinate.It has N- number pedestals on each cylindrical surface, which naturally forms the outer and inner cavity oil film gaps with radius of C1and C2.The angular positions θjand θiof the outer and inner pedestals differ by π/N, i.e., θi=β+2π（i-1)/N，θj=θi+π/N, where i， j are the location number of the inner and outer pedestals. Once the elastic ring is squeezed by the journal, bending deformation heproduces and be recorded from the starting angle α. This results in consist deformation among the inner chamber oil film gap hi, the outer chamber oil film gap ho,and the SFD oil film gap h2.Note that,for simplicity, the gap functions hi, ho, h2, and the deformation function he, are abbreviated from hi（θ，φ), ho（θ，φ), h2（θ，φ), and he（θ，φ) respectively, where θ is the angle of start of positive pressure region measured from the centerline and φ is the journal attitude angle.

An inertial Cartesian frame XOY is established through the bearing housing center O. Reynolds incompressible fluid is employed. Suppose that the rotor keeps running at a constant angular speed Ω such that the journal center OJorbits steadily about O with eccentricity e,i.e., ˙e=0， ˙φ=0.This combined squeezing and orbital motions generate positive oil film pressure p in both inner and outer cavities.Where if pressure difference occurs,fluids flow through the orifice at the net flow rate ud.

The steady-state governing equations for the oil film pressure in the inner and outer cavities, derived from the generalized Reynolds equation, can be described as23:

Fig. 1 Structural scheme of single-layer ERSFD.

To obtain definite pressure solutions from the elliptic partial differential Eqs. (1) and (2), the appropriate boundaries need to be determined in advance. It is mainly related to the oil-supply fashion, the end-seal type, and the starting boundary that periodically expands in the circumferential direction.For the entire ERSFD model in the present analysis,both ends are tightly sealed with sealing rings.When the elastic ring is no longer deformed and in a stable state, the variation of the oil film pressure along the axial direction is approximately zero.Hence, the pressure boundary condition at both ends is given by24–25

where psis the oil supply pressure,κ,?κ,and n-are the computational domain, the domain boundary and its corresponding normal vector outward the boundary,κ ?R2， ?κ:0 ≤θ ≤2π， 0 ≤z ≤l, where l is the damper length.

2.2. Thin-walled ring model

A thin-walled ring model is shown in Fig. 2. Assume that the ring is self-balanced under the applied load including journal acting force qiand outer pedestal reaction force Qj. For each ring portion separated by two outer pedestals, the force qihas identical circumferential angle π/N.The deformation at any section w-w with angle γ is estimated according to the curve deflection equation of thin-walled ring. The journal pushes the pedestal and contributes essential effect to the elastic ring deformation.10Therefore, the oil film force is omitted and assume that the force qidepends primarily on the journal motion and the ring stiffness.When the ring deforms,the total perimeter corresponding to the radius of curvature of the neutral layer of the elastic ring r remains unchanged. Boundary conditions of cross section at half ring, including bending moments M1,M2and tensile loads F1,F2,are dealt with using Castigliano’s theorem.

Fig. 2 Thin-walled ring model (orifices are not shown).

Acting force on the inner pedestal by the journal is represented by

where Fcis the component of unbalance force along the centerline, m is the rotor mass, a is the imbalance eccentricity of the rotor system,Fpis the radial component of the oil film force in the inner chamber,t is the time,and Fqis the radial component of the reaction force on the outer pedestal. Firstly, provide an initial value e such that qiis obtained by using Eq. (9), and then the oil film pressure p is solved following the procedures detailed in Section 2.4. Note that for any determined values Ω and φ, the force Fcis a constant. Finally, check the values Fc, Fp, and Fqusing Eq. (10) to tell whether the equilibrated relationship is met.If not satisfied,a new value e must be given and iterated again until qiis achieved.

The curve deflection equation of thin-walled ring is26:

where M is the sectional bending moment at the deformed section and EI is the flexural rigidity of the ring in the plane with initial curvature.It is noting that the variable γ can be replaced by θ by the relation of γ = hΔθ， h=1，2，3， ...， k/N, where k is the total number of divisions in the circumferential direction in finite difference scheme. Using the Fourier series expansion approach to represent the general solution of Eq. (11) gives

It is found in Eq.(14)and Eq.(15)that the deformation of the elastic ring heis not only influenced by the structural flexibility, λ=r3/4πEI, but also related to the spatial variable θ and temporal variable φ, respectively. Therefore, elastic ring deforms differently at each attitude angular position.

Fig. 3 Schematic diagram of turbulent jet model.

2.3. Turbulent jet model

The squeeze film flow in the oil chamber is generally considered to be stable and laminar because it enjoys a typical Reynolds numbers from one to twenty.However,whether the flow will develop into turbulence at this lower Reynolds numbers is still an open question.27Tichy28suggests that turbulent flow in SFD will occur when the Reynolds numbers exceeds 2000.Experimentally, the critical Reynolds number has been established equal to 1200.24Nevertheless, the flow through short hole is commonly taken as turbulent.29This means that the flow along the damping orifice of ERSFD has a Reynolds number not less than 1200. On the other hand, once the Reynolds number of the outlet flow is greater than 30,turbulent jet happens.30As the oil in the inner chamber is squeezed through the orifice into the outer chamber,the surrounding fluid of the outer cavity is continually entrained into the main flow,resulting in different outflow rate profiles along the jet trajectory.In order to accurately predict the net flow rate out of the orifice for ERSFD modeling, hence, a turbulent jet model is proposed.

Table 1 Dimensionless variables.

Fig. 4 Center finite difference scheme.

It manifests that the net outflow rate udis related to parameters such as damper length, journal radius, rotor rotational speed, squeeze film gap and elastic ring deformation. It has a nonlinear relationship with the number of orifices and their diameter.

2.4. Solving algorithm

Defining dimensionless parameters: Hi= hi/C1, Ho= ho/C1,He= he/C1, H2= h2/C1, P=p（C1/R)2/μΩ, L=2z/l,Ud=ud/C1Ω, and ε=e/C1, and assuming that C1=C2,rewrites variables in dimensionless form listed in Table 1.

Substituting Eq. (1) and Eq. (2)with dimensionless parameters,and then dividing both sides of the equations by the product of μΩC1yields

Fig. 5 Flowchart for solving oil film pressure.

Table 2 Simulation parameters.

Assuming the iterative accuracy obeys a relative error criterion,

where c, εeare the iterative number and iterative accuracy,respectively.

The flowchart to solve the oil film pressure is shown in Fig. 5. It involves two loop variables i， w, four independent unknown variables h2， he， hi, and ho, and three transitional variables qi, ?hi/?θ and ?he/?θ. When the loop variable i cycles, firstly, determining the acting load with Eq. (9), and then computing the gap function h2and elastic ring radial deformation heaccording to Eq.(3)and Eq.(14),respectively,finally, working out the derivative of ?he/?θ and ?hi/?θ in terms of Eq.(15)and Eq.(16),respectively.Once the loop variable w ends,arranging all the oil film gaps hiand hoto the first inner pedestal location, and then employing central finite difference Eq. (29), Eq. (30), Eq. (35), and Eq. (36) to solve Eq.(26)and Eq.(27)for oil film pressure under the error criterion Eq. (37).

3. Numerical analysis and experimental verification

Numerical analysis for the proposed model was implemented to examine the effect of elastic ring flexibility,orifice diameter,and attitude angular location on the OFPP. The simulation parameters are listed in Table 2 and the number of grid divisions in the finite difference algorithm is set to k=256,q=10.

The lubricating oil for the squeeze film is fed through an individual inlet located in the midspan of the ERSFD.In solving the pressure fields with numerical scheme, the oil supply pressure is treated in circumferential boundary conditions of the computational domain. When the journal is concentric with the bearing, the oil pressure both in outer and inner cavities is assumed equal to the oil supply pressure. In this case,the oil film flows consecutively along the axial direction, but discontinuously in the circumferential direction because of the barrier caused by the neighbor pedestals of the elastic ring.When the journal whirls about the bearing center with a certain orbital radius, it is assumed that the oil film is absolutely pressurized. Under this circumstance, the minimum pressure desired at the maximum oil gap is presumably not less than the supplied oil pressure. Finally, the supplied oil pressure is expressed in dimensionless form with the expression Ps=ps（C1/R)2/μΩ, and then substituted into the boundary condition Eq.(30)to solve the Eq.(26)and Eq.(27)for oil film pressure.

The assembly structure and trial bearing used in the test are in line with the actual in-service aero-engine. It is not allowed to damage the bearing and its accessories, leading to certain troubles in directly measuring the oil film pressure in the inner cavity. Therefore, this test contributes the verification to the proposed model by measuring the oil film pressure in the outer cavity. The investigation was conducted on a High Impact &Overload Integrated Bearing Fatigue Testing Machine. Fig. 6 shows the test rigs. The fundamental subsystem consists of the main test bench,lubricating system,hydraulic load system,frequency converter, cooler, and control system, as shown in Fig. 6(a). To isolate the vibration input and electromagnetic interference, flexible rope couplings and shielded cables are used.Fig.6(b)details a partially enlarged view of the main test bench to eye how the pressure transmitters are connected.Inside the main test bench, the distribution of pressure measurement points on the oil film bushing is shown in Fig. 6(d).A leakage outlet with a diameter of ?0.2 mm is equipped next to the end face of the ring to achieve necessary axial leakage for the squeeze oil film. The squeezed oil in the outer chamber is guided out of the machine body via the outlets of PMO1, PMO2, PMO3, and PMO4, and then connected to pressure transmitters (Model: HMIPH-10-A-0.2-AAAR) by using metal conduits. Finally, the detected pressure signal is fed to the synchronous acquisition card (Model: USB DAQ V5.2-DI), and recorded on the industrial computer (Model:ADVANTECH I.C. 510) using Lab Windows/CVI software.Fig.6(e)shows the fluid flow path inside the damper.The pedestals situate on the surface of the elastic ring in chessboard shape,forming a series of oil sub-chambers.The entire damper is tightly sealed at the end by radially placing compression seal ring into the seal groove. However, a short distance G exists between the seal groove and the top surface of the ring.Therefore, the outer cavity is not airtight such that the elastic ring deforms adaptively to adjust the oil gap between the inner and outer oil film chambers. The bearing radial load bends each of the ring parts located between the pedestals so that the oil is pressed from chamber to chamber through small orifices in the ring and tip clearances Λ,thus producing the damping effect. Fig. 6(c) shows the trial bearing. Except for 16 pedestals, the other parameters are the same as those in Table 2.

Fig. 6 Test rigs (PMO1-4: Pressure measuring outlets 1-4).

The squeeze film oozes from the leakage outlet at a low flow rate, further assuming that the flow remains unchanged.Therefore,the pressure keeps stable at four measuring outlets,as well as in the pipes. Within the first 15 minutes after the experiment started, the pressure transmitters were disconnected from the pipeline. This ensures that the air bubbles remaining in the oil film cavity are effectively removed, and the pressure in the pipe is fully balanced with that in the outer chamber. Consequently, there is almost no flow in the pipes,and because of the lower pressure loss, reliable and accurate measurement results can be obtained.

The main test processes are as follows: firstly, apply loads of 5000 N and 2000 N on the rotor system in axial and radial direction respectively.Meanwhile,feed aviation lubricating oil with a temperature and pressure of 40°C and 0.3 MPa,respectively, and then start the motorized spindle (Model:PM260SD22Q70)at 3000 r.min-1running at least 15 minutes to completely stabilize the system.Finally,the pressure signals were recorded with a sampling rate of 10,000 Hz at the speed of 6000 r.min-1, 8000 r.min-1, and 10,000 r.min-1. Each signal is 5000 samples long. For steady-state synchronous circular precession, the distribution of oil film pressure along the circumference of the journal is the same as the dynamic pressure change of a single fixed measuring point when the rotor rotates one revolution31. Therefore, only measurements from PMO1 channel were analyzed.

Fig. 7 Influence of elastic ring flexibility on the OFPP (λ = 5.0×10-6 mm. N-1).

4. Results and discussion

4.1. Influence of the elastic ring flexibility on OFPP

The oil film pressure was calculated with φ=π when the parameters of elastic ring flexibility were incrementally taken as: λ=5.0×10-6mm. N-1, λ = 3.0×10-5mm. N-1, and λ=1.0×10-4mm. N-1. The results are shown in Fig. 7 to Fig. 9. In order to make a comparison of the OFPP of the inner chamber with that of the SFD, substitute all terms on the right side of Eq. (26) with -12?Hi/?θ. Under the boundary conditions Eq. (30), Eq. (35), and Eq. (36), the oil film pressure of the SFD is solved assuming that no elastic ring deformation happens.

Fig. 8 Influence of elastic ring flexibility on the OFPP (λ = 3.0×10-5 mm. N-1).

Fig. 9 Influence of elastic ring flexibility on the OFPP (λ = 1.0×10-4 mm. N-1).

4.2. Influence of the orifice diameter on OFPP

The parametric analysis of the diameter of the orifice was implemented by decreasingly setting dc=1.2 mm,dc=1.0 mm and dc=0.6 mm when φ=π. The results are shown in Fig. 10.

Fig. 10 Influence of orifice diameter on the OFPP.

Fig. 11 Influence of typical attitude angular location on the OFPP (φ=π).

Fig. 12 Influence of successive attitude angular location on the OFPP (φ=2π/9 to φ=6π/9).

It is seen in Fig.10(a)that the oil film pressure in the outer cavity covers over an isolated domain and the maximum value at the position of φ=π is increased by nearly 4 times, corresponding to half of orifice diameter reduced. This may be because as the diameter of the orifice decreases, the resistance becomes stronger when the fluid flows through the orifice,leading to an increase in the oil film pressure on the elastic ring surface and further enhancement of the squeezing effect on the outer chamber. A fact has been shown that the transmitted force of the oil film in the outer cavity is increased by 2 times in the absence of damping orifice (equivalent to zero diameter of the orifice) but with an end seal.32

Fig. 13 Influence of successive attitude angular location on the OFPP (φ=8π/9 to φ=12π/9).

It is observed in Fig.10(b)that the maximum oil film pressure of the inner chamber is increased by close to 3 times as the orifice diameter is reduced by 50%.The expansion of the loadcarrying land promotes the fluid flow. This may be attributed to the fact that a smaller diameter of the orifice produces a higher damping force so that the load-carrying region is enlarged. It clearly shows that the oil film pressure both in outer and inner cavities is increased as the diameter of the orifice decreases.

4.3. Influence of attitude angular location on OFPP

The OFPP is examined considering a typical attitude angular location φ=π with the results shown in Fig. 11.

It is shown in Fig.11(a)and(b)that the oil film pressure in the outer cavity coats in an individual load-carrying land Ⅰand spreads symmetrically relative to the attitude angular location.A series of second spikes from s1to s8related to the inner pedestal location produces and the pressure profile changes consistently with the deformation of the elastic ring. Maximum pressure of Pmaxlies at the position corresponding to the utmost deformation of elastic ring He（max). It is seen in Fig. 11(c) and (d) that the oil film pressure in inner cavity extends to cover multiple contiguous regions and exhibits large convergence-oriented shift relative to the attitude angular location of φ=π.The pressure is more uniformly distributed with the highest pressure Pmaxlocating at the position corresponding to the minimum elastic ring deformation He（min). Some secondary pressure peaks associated with the outer pedestal locations (corresponding to marks G3, G4, and G5shown in Fig.11(d))also appears.It is noticed in Fig.11(b)and(d)that the oil film gap is discontinuous at the pedestals positions 1 to 8 and the oil film gap of the inner chamber becomes wider over the convergent zone due to the deformation of elastic ring.

To further look into the effect of the attitude angle on the OFPP,assume that the attitude angle varies from φ=2π/9 to φ=18π/9 with an incremental step of 2π/9, and yields the results shown in Figs. 12–14.

It is seen from the sub-graph (a) in Fig. 12 to Fig. 14 that the oil film pressure in outer cavity shifts synchronously and periodically as the attitude angle changes.The oil film pressure in inner cavity varies synchronously but not periodically with the attitude angular position, as shown in the sub-graph (b).It is found in the sub-graph(c)and(d)that the elastic ring segments are successively deformed with the variation of the attitude angular position, and no more than half of the ring segments deforms at each attitude angle position. The oil film gap of the outer cavity decreases but increases in the inner cavity at the deformed portion of the elastic ring. Note that the discontinuity of the oil film gap marked with G is caused by the fact that the movable attitude angle affects the oil film gap at the first inner pedestal position (the starting point used for recording the oil film gap).

Fig. 14 Influence of successive attitude angular location on the OFPP (φ=14π/9 to φ=18π/9).

It manifests that the oil film in outer cavity distributes symmetrically with respect to the attitude angular location and shifts synchronously and periodically, while the oil film pressure in the inner cavity spreads steadily towards the convergence zone relative to the attitude angular location and just varies synchronously.

Fig. 15 Comparison of the OFPP for nr =10000 r.min-1.

Fig. 16 Comparison of the OFPP for nr =8000 r.min-1.

4.4. Model validation

where dLis the Euclidean distance and dAis the cosine distance. According to Eq. (38) and Eq. (39), the distances are calculated respectively by using the model solutions and experimental data when the rotor runs at speeds from 6000 r.min-1to 10000 r.min-1. The result is shown in Fig. 19.

Fig. 17 Comparison of the OFPP for nr =6000 r.min-1.

Fig. 18 Comparison of the OFPP at three operating speeds.

It is seen in Fig. 15(a) that as the journal whirls with the rotor,the oil film pressure predicted with the model shifts synchronously and periodically. Around the pressure peak, a series of secondary pressure spikes related to the inner pedestal location clearly presents.It is found in Fig.15(b)that the measured pressure varies with the journal attitude angular position exhibiting similar behavior to the model, as well are emerged quite a few of secondary pressure spikes in the highestpressure region. Compared with the test results, the profile of oil film pressure obtained by the model presents some‘‘sharp” shapes, this may be responsible for the fact that the model does not allow for the influence of oil film pressure on the deformation of the elastic ring. It is seen in Fig. 16 and Fig. 17 that the same remark roughly holds for the Fig. 15.The oil film pressure distributes synchronously with the attitude angular location,except that the maximum pressure peak increases with the increase of rotor speed. It is observed in Fig. 18(a) that as the rotor runs at speeds incrementally from 6000 r.min-1to 10,000 r.min-1, the measured pressure has a slightly increase in amplitude as a whole, but the same periodic distribution relative to the attitude angular location,more obvious as seen in Fig. 18 (b).

It is seen in Fig. 19 that experimental and model results show the best fitting for higher cosine distance values at three speeds. Little is changed in the direction of two oil film pressure vectors.It illustrates that the model has an accordant tendency with the experimental results in terms of pressure distribution with the attitude angular location. Maximum relative error of the average oil-film pressure between the model and measured data is 1.13% happened at 6000 r.min-1. It suggests that the model is consistent with the test results in the amplitude variation of the oil film pressure. Furthermore,whether it is the model or experiment,the average oil film pressure increases with the increase of the rotor speed.It concludes that the model is well experimentally proved and capable of better pressure results.

5. Conclusions

Aiming at constructing an analytical model for a single-layer ERSFD, the thin-walled ring and turbulent-jet model is derived.The elastic ring radial deformation is obtained according to the thin-walled ring plane bending theory, and the orifice outflow rate is given by directly solving the Prandtl boundary layer differential equations.The central finite difference scheme is used to solve the steady-state oil film pressure.Numerical analysis was carried out to analyze the influence of elastic ring deformation,orifice diameter,and attitude angular location on the performance of oil film pressure. Experiment was performed to verify the effectiveness of the proposed model. The main conclusions achieved are as follows:

(1) The rise in elastic ring flexibility causes an increase in oil film pressure for the outer cavity but a decrease for the inner cavity. Low flexibility elastic ring affects the oil film pressure in inner cavity to behave a characteristic similar to the squeeze film damper.

(2) The decrease in the orifice diameter leads to an increase in oil film pressure for both the inner and outer cavities,and the small diameter orifice benefits the circumferential fluidity of the oil film in inner cavity.

(3) The almost balanced deformation of the elastic ring relative to the attitude angular location makes the oil film pressure in outer cavity symmetrically distribute over an individual load-carrying region, and that in inner cavity exhibit a large convergence-oriented shift with a uniform profile.

(4) The oil film pressure in outer cavity varies synchronously and periodically with the attitude angular location, but the amplitude increases slightly with the increase of rotor speed.The application of the proposed model on the ERSFD can better uncover the oil film pressure characteristics.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (No. 52005158).